The potential energy of a particle with displacement $X$ is $U(X)$. The motion is simple harmonic, when ($K$ is a positive constant)
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(a) $F = - \,kx$
==> $dW = Fdx = - \,kxdx$
So $\int_{\,0}^{\,W} {dW} = \int_{\,0}^{\,x} { - kx\,dx} $
==> $W = U = - \frac{1}{2}k{x^2}$
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